MEN OF MATHEMATICS Kronecker actually did this, at least in outline, and indicated how the whole of algebra and the theory of numbers, including algebraic numbers, can be reconstructed in accordance with his demand. To get rid of V — 1, for example, we need only put a letter for it temporarily, say i, and consider polynomials con- taining i and other letters, say x*y,z, ... Then we manipulate these polynomials as in elementary algebra, treating i like any of the other letters, till the last step, when every polynomial containing i is divided by z2 -j- 1 an<^ everything but the re- mainder obtained from this division is discarded. Anyone ^ho remembers a little elementary algebra may readily convince himself that this leads to all the familiar properties of the mysteriously misnamed ^imaginary' numbers of the text-books. In a similar manner negatives and fractions and all algebraic numbers (other than the positive rational integers) are elimi- nated from mathematics - if desired - and only the blessed positive integers remain. The inspiration about discarding V — 1 goes back to Cauchy in 1847. This was the germ of Kronecker" s programme. Those who dislike Kronecker's 'revolution' call it a Putsch, which is more like a drunken brawl than an orderly revolution. Nevertheless it has ltd in recent years to two constructively critical movements hi the whole of mathematics: the demand that a construction in a finite number of steps be given or proved to be possible for any 'number' or other mathematical 'entity' whose 'existence' is indicated, and the banishment from mathematics of all definitions that cannot be stated expli- citly in a finite number of words, insistence upon these demands has already done much to clarify our conception of the nature of mathematics, but a vast amount remains to be done. As this work is still in progress we shall defer further consideration of it until we come to Cantor, when it will be possible to exhibit examples. Kronecker* s disagreement with Weierstrass should not leave an unpleasant impression, as it may do if we ignore the rest of Kronecker's generous life. Kronecker had no intention of wounding his kindly old senior; he merely let his tongue run away with him in the heat of a purely mathematical argument, 532